420 XV NORMED ALGEBRAS AND SPECTRAL THEORY ing results then imply that (N - x \ N • y) = (N* -x\N*-y) for all x, y in dom(N)= dom(JV*) = F. This being so, if z e domC/WV*), then we have (N • x\ N- z) = (N* - x \N* - z) = (x | NN* • z) for all x e dom(A^). This proves that N • z e dom(JV*), by the definition of the adjoint (15.12.3), and that N*N - z = NN* - z. Thus we have established that dom(NN*) <=. dom(N*N) and that the operators NN* and N*N coincide on dom(7\W*). The proof of (ii) is now completed by interchanging the roles of AT and N* (since N** = N). Now return to the proof of (i). Consider the continuous hermitian operator B = (/ -f N*N)~"1 (15.12.6), whose spectrum is contained in I = [0, 1]. We shall first show that f(B)(dom(NJ) c domCAO and f(B)N - x = Nf(B) • x for all/6 ^C(I) and all x e dom(N). First of all take/= lc; if x e dom(AO, we may write BN'X-= BN(I + N*N)B * x = JS(N + NN*N)B • x, recalling that NB(E) c dom(A^*) and N*NB -x = x-B-xe dom(N), be- cause fl(E) c dom(^) (15.12.6). But N*N = NN*, hence BN*x=* B(N + N*NN)B * x = B(1 4- N*N)NB -x = NB-x; that is, fl/V" • ^r = 7V# • A: for all x e dom(N). By induction on n, we obtain J"(dom(JV)) a dom(N) &nd BnN - x = NBn • x for all jc e dom(N) and all integers n > 0. This establishes our assertion when /is a polynomial with real coefficients. Hence, for such a polynomial / we have (15.12.8.2) (f(B)N -x \y) = (/(£) • x | N* • y) for all x e dom(AO and j e dom(AO. Having regard to (15.11.2) applied to B, we deduce that J/(0 dm*.,.,(0 = J /(0 dm,,^.,(0, and since the restrictions of polynomials with real coefficients to Sp(2?) are dense in ^R(Sp(J)) by the Weierstrass approximation theorem (7.4.1 ), the formula above shows that mN.Xty = mX)N*.y ((3.15.1) and (13.2)). Conse- quently, by applying (15.11.2) to B, the formula (15.112,8.) is valid for all functions /e^c(I). But this formula shows that f(B) • x e dom(7V**) = dom(AT) and that (f(B)N -x\y) = (Nf(B) -x\y) for all y e dom(N). Since dom(AT) is dense in E, it follows that Nf(B) 'X=f(B)N-x for all x e dom(N). n